I have seen the usage of the word kio used as a question word "what" a lot of times:
Kio estas tio? (what's this?)
But lately I have seen this form:
Kion vi manĝas? (what are you eating?)
And I was confused. If "kio" can be used as an object of a sentence like the above example, can it be the object of a sentence which is not a question either (like in English)?
Tio estas kion mi vidis! (This is what I saw)
Yes, you can use "kion" to introduce a relative sub-clause. Your sentence "Tio estas kion mi vidis!" is correct.
The interesting thing about this example sentence is that it touches a detail of Esperanto grammar that is not well documented in our most complete grammar resource PMEG. Let me explain.
A grammatically more complete way of saying "That is what I saw" would be "Tio estas tio, kion mi vidis." Now in constructions of the form "tio(n), kio(n)", one has the option to drop "tio(n)". PMEG says that normally dropping "tio(n)" is only permitted when it is in the same case as "kio(n)" (i.e. if both have the -n or both lack it).
However, PMEG also acknowledges that there are some contexts in which "tio" is commonly dropped from "tio, kion", but PMEG does not explain in detail in which contexts this is permissible. It only gives the example of sentences with "jen":
Jen [estas] tio, kion diris Aĥitofel! → Jen kion diris Aĥitofel!
By searching for the patterns "tio estas kion" and "tio estas tio kion" in the Tekstaro de Esperanto (an Esperanto corpus consisting of texts by well considered authors, inclusing many texts by Zamenhof), I realized that in such a context too, "tio" is commonly dropped despite being in a different case than the following "kion". Already Zamenhof himself dropped "tio" in such a context.
I will suggest to Bertilo Wennergren, the author of PMEG, to mention this context as another example, in which "tio" can be dropped before "kion", so as to make his explanation more complete (he has already previously made changes to PMEG based on my suggestions).
Yes, that is correct, you can use either kion or kio to introduce a sub-phrase. Which one to use depends on whether the thing it is referencing is being used as the subject or the object in the sub-phrase (not in the main phrase). For example:
Tio estas kion mi vidis!
(In this case the sub-phrase references the tio from the main phrase as the object)
Tio estas kio okazis
(In this case it is the subject)
Also note that using kio or kion is particular to this example and in some cases it is necessary to use kiu / kiun instead. Kio is only used in this case because it is referring to another correlative which also ends in o. In this case the correlative refers to tio so it must match and be kio. If the thing being referred to is a normal noun you would use kiu or kiun instead:
La pomo kiun mi manĝis estis bona.
or:
La pomo kiu estis en la fridujo estis bona.
As always, there is a more detailed explanation of this in the grammar section of Lernu.
[I now consider this old answer of mine incorrect. Please see my newer answer.]
To say "This is what I saw" in Esperanto, you have to say "Tio estas tio, kion mi vidis." In constructions of the form "tio(n), kio(n)", one has the option to drop "tio(n)", but only if it is in the same case as "kio(n)" (i.e. if both have the -n or both lack it). So you cannot drop "tio" in "tio, kion".
This is explained in some more detail and with some examples in PMEG.
